Question

All Seasons Plumbing has two service trucks that frequently need repair. If the probability the first truck is available is .75, the probability the second truck is available is .50, and the probability that both trucks are available is .30, what is the probability neither truck is available

Answer 1

5% probability neither truck is available

Explanation:

We solve this problem building the Venn's diagram of these probabilities.

I am going to say that:

A is the probability that the first truck is available.

B is the probability that the second truck is available.

We have that:

`A = a + (A \cap B)`

In which a is the probability that the first truck is available and the second one is not
`A \cap B` is the probability that both trucks are available.

By the same logic, we have that:

`B = b + (A \cap B)`

The probability that both trucks are available is .30.

This means that
`A \cap B = 0.3`

The probability the second truck is available is .50

This means that
`B = 0.5`. So

`B = b + (A \cap B)`

`0.5 = b + 0.3`

`b = 0.2`

The probability the first truck is available is .75

This means that
`A = 0.75`. So

`A = a + (A \cap B)`

`0.75 = a + 0.3`

`a = 0.45`

The probability that at least one truck is available is:

`A \cup B = a + b + (A \cap B) = 0.45 + 0.2 + 0.3 = 0.95`

The probability that neither truck is available is:

`1 - (A \cup B) = 1 - 0.95 = 0.05`

5% probability neither truck is available